Theorem dfsucon 44111

Description: 𝐴 is called a successor ordinal if it is not a limit ordinal and not the empty set. (Contributed by RP, 11-Nov-2023.)

Assertion

Ref	Expression
dfsucon	⊢ ((Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅) ↔ ∃𝑥 ∈ On 𝐴 = suc 𝑥)

Distinct variable group:   𝑥,𝐴

Proof of Theorem dfsucon

Step	Hyp	Ref	Expression
1		3ancomb 1114	. . . 4 ⊢ ((Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅) ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴))
2		df-3an 1103	. . . 4 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) ↔ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) ∧ ¬ Lim 𝐴))
3		df-ne 2961	. . . . . . . . 9 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
4	3	anbi2i 634	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) ↔ (Ord 𝐴 ∧ ¬ 𝐴 = ∅))
5	4	imbi1i 352	. . . . . . 7 ⊢ (((Ord 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ ((Ord 𝐴 ∧ ¬ 𝐴 = ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥))
6		pm5.6 1017	. . . . . . 7 ⊢ (((Ord 𝐴 ∧ ¬ 𝐴 = ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (Ord 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
7		iman 406	. . . . . . 7 ⊢ ((Ord 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)) ↔ ¬ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
8	5, 6, 7	3bitrri 301	. . . . . 6 ⊢ (¬ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)) ↔ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥))
9		dflim3 7831	. . . . . 6 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
10	8, 9	xchnxbir 336	. . . . 5 ⊢ (¬ Lim 𝐴 ↔ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥))
11	10	anbi2i 634	. . . 4 ⊢ (((Ord 𝐴 ∧ 𝐴 ≠ ∅) ∧ ¬ Lim 𝐴) ↔ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) ∧ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
12	1, 2, 11	3bitri 300	. . 3 ⊢ ((Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅) ↔ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) ∧ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
13		pm3.35 814	. . 3 ⊢ (((Ord 𝐴 ∧ 𝐴 ≠ ∅) ∧ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)
14	12, 13	sylbi 220	. 2 ⊢ ((Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)
15		eloni 6360	. . . . . 6 ⊢ (𝑥 ∈ On → Ord 𝑥)
16		ordsuc 7798	. . . . . 6 ⊢ (Ord 𝑥 ↔ Ord suc 𝑥)
17	15, 16	sylib 221	. . . . 5 ⊢ (𝑥 ∈ On → Ord suc 𝑥)
18		nlimsuc 44029	. . . . 5 ⊢ (𝑥 ∈ On → ¬ Lim suc 𝑥)
19		nsuceq0 6435	. . . . . 6 ⊢ suc 𝑥 ≠ ∅
20	19	a1i 11	. . . . 5 ⊢ (𝑥 ∈ On → suc 𝑥 ≠ ∅)
21	17, 18, 20	3jca 1144	. . . 4 ⊢ (𝑥 ∈ On → (Ord suc 𝑥 ∧ ¬ Lim suc 𝑥 ∧ suc 𝑥 ≠ ∅))
22		ordeq 6357	. . . . 5 ⊢ (𝐴 = suc 𝑥 → (Ord 𝐴 ↔ Ord suc 𝑥))
23		limeq 6362	. . . . . 6 ⊢ (𝐴 = suc 𝑥 → (Lim 𝐴 ↔ Lim suc 𝑥))
24	23	notbid 321	. . . . 5 ⊢ (𝐴 = suc 𝑥 → (¬ Lim 𝐴 ↔ ¬ Lim suc 𝑥))
25		neeq1 3022	. . . . 5 ⊢ (𝐴 = suc 𝑥 → (𝐴 ≠ ∅ ↔ suc 𝑥 ≠ ∅))
26	22, 24, 25	3anbi123d 1460	. . . 4 ⊢ (𝐴 = suc 𝑥 → ((Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅) ↔ (Ord suc 𝑥 ∧ ¬ Lim suc 𝑥 ∧ suc 𝑥 ≠ ∅)))
27	21, 26	syl5ibrcom 250	. . 3 ⊢ (𝑥 ∈ On → (𝐴 = suc 𝑥 → (Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅)))
28	27	rexlimiv 3159	. 2 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅))
29	14, 28	impbii 212	1 ⊢ ((Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅) ↔ ∃𝑥 ∈ On 𝐴 = suc 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ∃wrex 3089  ∅c0 4288  Ord word 6349  Oncon0 6350  Lim wlim 6351  suc csuc 6352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-tr 5213  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356

This theorem is referenced by: (None)